 An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation ProblemJournal of Optimization Theory and Applications, 2001, vol. 110, issue 2, No 10, 429-443 Abstract: Abstract We prove the following theorem. Let m and n be any positive integers with m≤n, and let $$T^n = \{ x \in \mathbb{R}^n |\Sigma _{i = 1}^n x_i = 1\}$$ be a subset of the n-dimensional Euclidean space ℝ n . For each i=1, . . . , m, there is a class $$\{ M_i^j {\text{| }}j = 1,...,n\}$$ of subsets M i j of Tn . Assume that $$\cup _{j = 1}^n M_i^j = T^n ,$$ for each i=1, . . . , m, that M i j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that $$x \in T^n$$ and its jth component xj≤B(i, j) imply $$x\not \in M_i^j$$ . Then, there exists a partition $$(\Pi (1),...,\Pi (m))$$ of {1, . . . , n} such that $$\Pi (i) \ne \emptyset$$ for all i and $$\cap _{i = 1}^m \cap _{j \in \Pi (i)} M_i^j \ne \emptyset .$$ We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem. Keywords: Intersection theorem; combinatorial theorem; fair allocation (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:110:y:2001:i:2:d:10.1023_a:1017587615488 Ordering information: This journal article can be ordered from Access Statistics for this article Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull More articles in Journal of Optimization Theory and Applications from Springer |
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